Hamiltonian-type elliptic system


Here is an example of Hamiltonian elliptic systems, also called the Lane-Emden system,

\begin{displaymath} (LE) \; \left\{ \begin{array}{ll} - \Delta u = v^p & \, x \... ...u=v=0 & \, x \in \partial \Omega, \nonumber \end{array}\right. \end{displaymath}


where $p,q>0,\Omega\subset {\mathbb{R}}^N (N\ge 1)$ is an open bounded domain. (LE) is called sublinear (superlinear) if $pq<1 \; (pq>1)$. The associated energy functional to (LE) is

\begin{displaymath} J(u,v)=\int_{\Omega}\nabla u \nabla v dx - \int_{\Omega}(\frac{1}{q+1}u^{q+1}+\frac{1}{p+1}v^{p+1})dx. \end{displaymath}



Some numerical results for (LE) on a disk (N=2): p=0.3, q=0.6 (sublinear case); p=5, q=3 (superlinear case).

Moreover, if letting $q=1$ and replacing $v^p$ with a general nonlinear function $f(x,v)$, then (LE) becomes 

a semilinear biharmonic problem with so-called Navier boundary conditions, see also biharmonic. 




Created by Xianjin Chen

Feb-28-2008